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The number of employees in a nationalised bank in a small town is 10, out of which 4 are female and the rest male. A committee of 5 is to be formed. If m be the number of ways to form such a committee in which there is atleast one female employee and n be the number of ways to form such a committee whichincludes at least two male employees, then find the ratio m:n.
  • a)
    3:2
  • b)
    5:1
  • c)
    1:1
  • d)
    0.8:0.9
Correct answer is option 'C'. Can you explain this answer?
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The number of employees in a nationalised bank in a small town is 10, ...
Now take different cases to fulfill the conditions given.
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The number of employees in a nationalised bank in a small town is 10, ...
Total Number of ways to select a committee of 5 from 10 employes = 10C5

m = At least one female employee = Total ways to select committee - Total ways of selecting all male employees

hence m = 10C5 - 6C5



n = At least 2 male employee  = Total ways to select committee  - the case of selecting 4 female and one male employee

hence n = 10C5 - (4C4 * 6C1) = 10C5 -6C1 = 10C5 - 6C5


hence  m:n = 1:1

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The number of employees in a nationalised bank in a small town is 10, ...
To find the ratio m:n, we need to first calculate the number of ways to form a committee with at least one female employee (m) and the number of ways to form a committee with at least two male employees (n).

Number of ways to form a committee with at least one female employee (m):
There are 4 female employees in the bank, and we need to choose at least one of them for the committee. The remaining committee members can be chosen from the remaining employees (both male and female). Therefore, we have the following possibilities:
1 female employee and 4 other employees (either male or female) - 4C1 * 6C4 = 4 * 15 = 60 possibilities
2 female employees and 3 other employees - 4C2 * 6C3 = 6 * 20 = 120 possibilities
3 female employees and 2 other employees - 4C3 * 6C2 = 4 * 15 = 60 possibilities
4 female employees and 1 other employee - 4C4 * 6C1 = 1 * 6 = 6 possibilities

Total number of possibilities (m) = 60 + 120 + 60 + 6 = 246

Number of ways to form a committee with at least two male employees (n):
There are 6 male employees in the bank, and we need to choose at least two of them for the committee. The remaining committee members can be chosen from the remaining employees (both male and female). Therefore, we have the following possibilities:
2 male employees and 3 other employees (either male or female) - 6C2 * 8C3 = 15 * 56 = 840 possibilities
3 male employees and 2 other employees - 6C3 * 8C2 = 20 * 28 = 560 possibilities
4 male employees and 1 other employee - 6C4 * 8C1 = 15 * 8 = 120 possibilities
5 male employees - 6C5 = 6 possibilities

Total number of possibilities (n) = 840 + 560 + 120 + 6 = 1526

Therefore, the ratio m:n = 246:1526, which can be simplified to 1:6. The correct answer is option 'C' (1:1).
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The number of employees in a nationalised bank in a small town is 10, out of which 4 are female and the rest male. A committee of 5 is to be formed. If m be the number of ways to form such a committee in which there is atleast one female employee and n be the number of ways to form such a committee whichincludes at least two male employees, then find the ratio m:n.a)3:2b)5:1c)1:1d)0.8:0.9Correct answer is option 'C'. Can you explain this answer?
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